Intersection (set theory)

Intersection (set theory)
In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to
Notation and terminology
Intersection is written using the symbol "" between the terms; that is, in infix notation. For example:
The intersection of more than two sets (generalized intersection) can be written as:
which is similar to capital-sigma notation.
For an explanation of the symbols used in this article, refer to the table of mathematical symbols.
Definition
The intersection of two sets and denoted by , is the set of all objects that are members of both the sets and
In symbols:
That is, is an element of the intersection if and only if is both an element of and an element of
For example:
The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
The number 9 is in the intersection of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of odd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.
Intersecting and disjoint sets
We say that if there exists some that is an element of both and in which case we also say that . Equivalently, intersects if their intersection is an , meaning that there exists some such that
We say that if does not intersect In plain language, they have no elements in common. and are disjoint if their intersection is empty, denoted
For example, the sets and are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.
Algebraic properties
Binary intersection is an associative operation; that is, for any sets and one has
Thus the parentheses may be omitted without ambiguity: either of the above can be written as . Intersection is also commutative. That is, for any and one has
The intersection of any set with the empty set results in the empty set; that is, that for any set ,
Also, the intersection operation is idempotent; that is, any set satisfies that . All these properties follow from analogous facts about logical conjunction.
Intersection distributes over union and union distributes over intersection. That is, for any sets and one has
Inside a universe one may define the complement of to be the set of all elements of not in Furthermore, the intersection of and may be written as the complement of the union of their complements, derived easily from De Morgan's laws:
Arbitrary intersections
The most general notion is the intersection of an arbitrary collection of sets.
If is a nonempty set whose elements are themselves sets, then is an element of the of if and only if for every element of is an element of
In symbols:
The notation for this last concept can vary considerably. Set theorists will sometimes write "", while others will instead write "".
The latter notation can be generalized to "", which refers to the intersection of the collection
Here is a nonempty set, and is a set for every
In the case that the index set is the set of natural numbers, notation analogous to that of an infinite product may be seen:
When formatting is difficult, this can also be written "". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.
Nullary intersection
Note that in the previous section, we excluded the case where was the empty set. The reason is as follows: The intersection of the collection is defined as the set (see set-builder notation)
If is empty, there are no sets in so the question becomes "which 's satisfy the stated condition?" The answer seems to be . When is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection),
but in standard (ZF) set theory, the universal set does not exist.
However, when restricted to the context of subsets of a given fixed set , the notion of the intersection of an empty collection of subsets of is well-defined. In that case, if is empty, its intersection is . Since all vacuously satisfy the required condition, the intersection of the empty collection of subsets of is all of In formulas, This matches the intuition that as collections of subsets become smaller, their respective intersections become larger; in the extreme case, the empty collection has an intersection equal to the whole underlying set.
Also, in type theory is of a prescribed type so the intersection is understood to be of type (the type of sets whose elements are in ), and we can define to be the universal set of (the set whose elements are exactly all terms of type ).
See also
References
Further reading
External links
Basic concepts in set theory
Operations on sets
Intersection